# Intersection of pullback divisor and exceptional divisor

Let $$X,Y$$ are nonsingular complex projective surfaces and $$f:Y\to X$$ be a birational morphism. Let $$E_1,\ldots,E_k$$ are irreducible exceptional divisors of $$f$$.

In this situation, we have $$K_Y=f^{\ast}K_X+\sum_{i=1}^{k}m_iE_i (m_i>0)$$ ($$K_Y,K_X$$: canonical divisor of $$Y,X$$)

How to prove $$(f^{\ast}K_X)\cdot E_i=0 (\forall i)$$?

More generally,Is it true that $$(f^\ast D)\cdot E_i=0$$ for all $$i$$ and for all divisor $$D$$ on $$X$$?

• What are $f(E_i)$? Can you say that if $D$ is any divisor on $X$, it is linearly equivalent to a divisor disjoint from $f(E_i)$? – Mohan May 7 at 17:45